Module 9 â Reorganized - eusddata

S D 

C S 

SAN DIEGO CITY SCHOOLS 

Instructional Module to Enhance the Teaching of 

H A R C O U R T 

Math 

California Edition 

Grade 5 

Module 9 − Reorganized 

OPERATIONS WITH FRACTIONS: 

MULTIPLICATION AND DIVISION 

Reorganized 8/20/04

Harcourt Math−Grade 5 MODULE 9 

⎯ WORK IN PROGRESS 

Revised 2/09/2 2


Revised 2/09/3 3


Grade Five Traditional Calendar – 2004-2005 

Order of Units and Pacing Guide 

Month Module Number of Days 

Module 1: Data and Graphing 

11 days 

September 

19 instructional days 

Module 2: 

Place Value and Addition and 

Subtraction of Whole Numbers and 

Decimals 

8 days 

October 





Decimals 

7 days 


Algebra: Use Addition and 

Multiplication; Integers 

14 days 

November 





8 days 


Geometry 

10 days 

December 


Winter Break 12/20 – 12/31 

January 





Geometry 

Multiply Whole Numbers and 

Decimals; Percent 



2 days 

11 days 

3 days 


Divide Whole Numbers and Decimals 

17 days 

February 



Number Theory; Fraction Concepts 

and Addition and Subtraction of 

Fractions 

17 days 

March 


Spring Break 3/21 – 3/25 


Number Theory; Fraction Concepts 

and Addition and Subtraction of 

Fractions 

5 days 


Geometry: Area, Perimeter, and 

Volume 

12 days 

April 


May 


STAR 4/26 – 5/17 

DMT 5/9 – 5/13 

June 



Operations with Fractions: 

Multiplication and Division 

Module 10: Measurement, Probability and Ratio 

Module 10 

Measurement, Probability and Ratio 

Review Grade 5 concepts 

Preview Grade 6 Concepts 

11 days 

9 days 

12 days 

13 days 

Revised 2/09/4 4


Grade Five – Year Round Calendar – 2004-2005 

Order of Units and Pacing Guide 

Month Module Number of Days 

September Module 1: Data and Graphing 

11 days 

19 instructional days Module 2: Place Value and Addition and 


Decimals 

8 days 

October 






Decimals 



7 days 

14 days 

November 






Geometry 

8 days 

10 days 

December 


Winter Break 12/22 – 1/17 



Geometry 



2 days 

11 days 

January 




Multiply Whole Numbers and Decimals; 

Percent 


3 days 

7days 

February 





Number Theory; Fraction Concepts and 

Addition and Subtraction of Fractions 

10 days 

7 days 

March 


Spring Break 3/16 – 4/8 

April 


May 












Geometry: Area, Perimeter, and Volume 

Geometry: Area, Perimeter, and Volume 

Operations with Fractions: Multiplication 

and Division 

Measurement, Probability and Ratio 

11 days 

4 days 

11 days 

1 days 

11 days 

9 days 

June 


STAR 5/27 – 6/17 

DMT 6/13 – 6/17 

July 


Module 10: Measurement, Probability and Ratio 12 days 

Review Grade 5 concepts 

Preview Grade 6 Concepts 

14 days 

Revised 2/09/5 5

San Diego City Schools 

Instruction and Curriculum Division 

MATHEMATICS CURRICULUM MAP – GRADE 5 

MODULE 9 – OPERATIONS WITH FRACTIONS: MULTIPLICATION AND DIVISION 

Modules represent individual units of study that lead to essential learnings 

THREADS THROUGHOUT THE YEAR: 

The threads represent ongoing learning opportunities in which students should be actively engaged throughout all units of inquiry during the entire 

school year. These items should not be isolated to any one particular unit of inquiry. 

Students will: 

1. Develop understanding of numbers and the number system and use their understanding to solve problems and recognize reasonable results. 

2. Develop understanding of and fluency in basic computation and procedural skills. 

3. Use mathematical reasoning to solve problems. 

4. Communicate their mathematical thinking using words, numbers, symbols, graphs and charts, and translate between the different representations. 

5. Use equations and variables to express generalizations of patterns and relationships. 

6. Develop logical thinking to analyze evidence and build arguments to support or refute a hypothesis. 

7. Make connections among mathematical ideas and between other disciplines 

8. Develop and use strategies, skills, and concepts to solve problems. 

9. Use appropriate tools, including technology, as vehicles to learn mathematical concepts. 

These are essential learnings that represent bigger ideas/concepts: 

• Students use their understanding about the properties and meanings of operations for whole numbers for estimation and computation with 

fractions. 

• Students use their understanding about relationships between operations of whole numbers for estimation and computation with fractions. 

• Students use their fraction sense for estimation and computation with fractions. 

• When dividing by fractions, students understand that the two ways of thinking about division--partition and measurement--result in two 

different division procedures.. 

These are essential questions that learners ask themselves in order to achieve the essential learnings: 

• *How do I use my understanding of basic operations with whole numbers to make meaning of multiplication and division with fractions? 

• How do I use models, pictures, and manipulatives to understand and solve multiplication problems with fractions? 

• How does the use of the *distributive property and the area model for multiplication of whole numbers connect to using the area model for multiplication of 

fractions? 

• How do I use the *inverse relationship between multiplication and division of whole numbers to help me understand and solve multiplication and division of fraction 

problems? 

* Presented in previous grade(s) 

Resources: Van de Walle, Chapter 16 (pp. 270-278); Mathematics Source Book, (pp.59-70) 

Revised 2/09/04 6


Revised 8/20/04 7

Key Mathematical Concepts: 

Harcourt Math – Grade 5 

Operation with Fractions: Multiplication and Division 

Chapters 21, and 22 

MODULE 9 

• The word “of” as used in “ 1/4 of 2/3 “ indicates multiplication. 

• Division of fractions asks the same question asked in division of whole numbers: 

12 ÷ 4 asks “How many 4’s are in 12?” 

1 1/2 ÷ 1/4 asks “How many 1/4’s are in 1 1/2?” 

• Multiplication and division are inverse operations. This means that dividing by a whole 

number or fraction has the same answer as multiplying by the reciprocal of the divisor. 

Revised 2/09/04 8


Harcourt Mathematics 

Grade 5 

Module 9: Operations with Fractions: Multiplication and Division 

Chapters 21, and 22 

MODULE 9 NOTES 

• The vocabulary used in this module can be difficult all learners, especially English 

Language Learners. Reinforce fraction vocabulary with a chart as suggested on p. 

344B, “English Language Learners”. 

• Extra time is spent developing the concept of multiplication of fractions (21.2 – two 

lessons; 21.2 – two lessons) before introducing the algorithm. (Number Sense 2.4) 

• Several models are provided as tools to think about division of fractions beyond just 

using the reciprocal. (Number Sense 2.4) 

• Lesson plans were not provided for problem-solving lessons 21.5, and 22. These 

lessons do not focus on the Key Standards for the grade level. (Mathematical 

Reasoning 2.1 – estimation -- is emphasized in this unit.) Writing their own problems 

deepens student understanding of concepts and enhances their ability to choose the 

correct operation for a given problem. 

Revised 8/20/04 9


Operations with Fractions: Multiplication and Division 

11 Days of Instruction: Chapters 21, and 22 

Day 1 

CHAPTER 21 

Multiply 

Fractions 

Lesson 21.1 

Multiply Fractions 

& Whole 

Numbers, Part 1 

Day 2 

Lesson 21.1 


& Whole 

Numbers, Part 2 

Day 3 

Lesson 21.2 

Multiply a Fraction 

by a Fraction, Part 

1 

Day 4 

Lesson 21.2 

Multiply a Fraction 

by a Fraction, Part 

2 

Day 5 

Lesson 21.3 


& Mixed Numbers 

Day 6 

Day 7 

Day 8 

Day 9 

Day 10 

Lesson 21.4 

Multiply with 

Mixed Numbers 

Day 11 

CHAPTER 22 

Divide Fractions 

Lesson 22.1 

Hands On: Divide 

Fractions 

Lesson 22.2 

Reciprocals 

Lesson 22.3 

Divide Whole 

Numbers by 

Fractions 

Lesson 22.4 


Assessment 

Chapters 21 & 22 

Revised 8/20/04 10

DAY: 1 

Unit 6: Operations with Fractions 

Module 7: Chapter 21: Multiply Fractions 

LESSON 21.1, Pp. 380-381 Part 1 

MATERIALS: 

LESSON FOCUS: 

CALIFORNIA 

STANDARDS: 

PURPOSE OF 

LESSON: 

LAUNCH: 

Introduce students 

to concepts. 

Books closed. 

One copy of TR 19 and 20 for each student and overhead transparencies 

for the teacher (or have students draw the open number lines); Challenge 

21.1 for homework if you choose that assignment. 

Fraction Bars for each student. 

Multiply Fractions and Whole Numbers 

Number Sense 2.4: Understand the concept of multiplication and division of 

fractions; 

2.5 Compute and perform simple multiplication and division fractions and 

apply these procedures to solving problems. 

• To use models (Fraction Kit/Fraction Bars, circles, arrays, number lines) to 

help students understand the concept of multiplication of fractions; 

• To understand the word “of” as in 3 groups of 3/4 (3 × 3/4) indicates 

multiplication; 

• To connect multiplication with fractions to multiplication of whole 

numbers/repeated addition. 

Alternative Teaching Strategy, P. 380B. Repeat with different examples. 

Learn, P. 380: Baker’s Dozen. 

• Write problem on board/overhead. 

• Ask them to discuss the problem with a partner. 

• Students work in pairs to solve the problem. Use models. Share solutions 

and strategies. 

• Ask students how they might write the problem: 

- With words: three groups of three-fourths. 

- As repeated addition: 3/4 + 3/4 + 3/4. 

- As multiplication: 3 × 3/4 (the “of” indicates multiplication). 

Teach, P. 380; Guided Instruction questions to guide discussion. 

• Ask students to think about how to model the problem with the number line 

labeled fourths on TR 19. (Circle/highlight 3 groups of 3 fourths from left to 

right as shown below.) 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

0 

1 

2 

1 

1 

2 

2 

1 

2 

3 

Revised 8/20/04 11


Discuss Step 3: 

How many wholes in 9/4? 

• Students may use their Fraction Kit/Fraction Bars, the open number 

line on TR 19, and any procedure they have learned to answer. 

• Ask students for a procedure for changing an improper fraction to a 

mixed number to explain to the class why it works. 

EXPLORE: 

Work with the 

concept. Focus 

on students 

“doing” 

mathematics. 

PRACTICE: 

Focus on 

Communication 

and 

Representation. 

SUMMARIZE: 

Connect purpose 

to activities. 

Check, P. 381 #1. 

• Discuss. 

• Then do #2 – 6. Work with a partner. 

• Draw a picture. 

• Ask students to work with a partner to solve them using TR 19. Ask them 

to note any patterns or shortcuts they notice as they work. Discuss. 

Practice & Problem Solving, P. 381 # 10-12. 

Then, #23 – 26. Use fraction model or number line. 

ASSESS, P. 381: DISCUSS. 

Share. 

• Ask students to discuss any patterns of shortcuts they noticed as they 

worked. 

o Multiply whole number by the numerator to get the number of 

parts 3 × 2/3 = 6 thirds 

o The denominator does not change 3 × 2/3 = 6/3 (Ask why.) 

o You can skip count 2-4-6 thirds 

HOMEWORK: Challenge 21.1 

• Write the number sentence for each picture. 

ROUTINES: 

Mixed Review, P. 381. 

Students needing more practice changing improper fractions to 

mixed numbers and mixed numbers to improper fractions will benefit from 

using TR 19 and 20 to practice those skills. 

Students can draw open number lines to model multiplication of a fraction 

by a whole number. They can write “story” problems to go with each 

problem. These could be copied on construction paper to post for the class. 

Revised 3/05/04 12


DAY: 2 



LESSON 21.1, Pp. 380-381 Part 2 

MATERIALS: 

LESSON 

FOCUS: 

CALIFORNIA 

STANDARDS: 

PURPOSE OF 

LESSON: 

LAUNCH: 

Introduce 

students to 

concepts. 

Overhead transparency of Reteach 21.1 for the teacher and one copy per 

student if the class can’t work from the overhead; 

one copy of Practice 21.1 per student for homework; 

Fraction Kit/Fraction Bars 

Multiply Fractions by a Whole Number 

Number Sense 2.4: Understand the concept of multiplication and division of 

fractions; 

2.5 Compute and perform simple multiplication and division fractions and 

apply these procedures to solving problems. 

• To use models and pictures to help students understand how to find a 

fractional part of a whole number ( 2/3 × 12); 

• To understand that the product of a whole number and a fraction less than 

one is less than the whole number. 

• To connect to multiplying by decimals (.2 × 10), and finding the percent of a 

number (25% of 16) 

• Show overhead transparency of the top part of Reteach 21.1. (Or draw on 

the board.) 

• Discuss how the problem was represented with a picture. 

• How many did you start with? (12) 

• How many groups? (3) thirds 

• How many thirds? (2) 

• Complete page and discuss. 

Warm-up: 

There are 12 students at the homework center. One third of them are boys. 

How many students at the homework center are boys? 

• Students work in pairs to solve and share their strategies. 

• Ask students to draw a picture of the problem. Use number lines or other 

models. (Accept student suggestions- maybe 12 faces, stick figures, or 

other.) 

What fraction of the group are boys? (1/3) How many parts/equal groups 

do we need to make? (3) Circle three equal groups. 

Possible Prompts: 

What does this problem ask us to find? 

Ask: How can I write it in words? (one-third of twelve) 

Ask: How can I write a number sentence for this problem? 

1/3 × 12 = ? 

Revised 3/05/04 13


EXPLORE: 

Work with the 


on students 

“doing” 

mathematics. 

PRACTICE: 

Focus on 

Communication 

and 


SUMMARIZE: 

Connect 

purpose to 

activities. 

Practice & Problem Solving, P. 381 #27 – 32. 

• Write with partners. Draw/use models. Discuss. 

Practice & Problem Solving, P. 381 # 13,14,15,22, p. 381 

ASSESS, P. 381: WRITE. 

HOMEWORK: Practice 21.1 

• Discuss Common Error Alert, p. 381TE margin. 

ROUTINES: 

Double number lines are another model students can use to find fractional 

parts of whole numbers and connect that work to work with percents. 

0 1/4 1/2 3/4 1 

0 3 6 9 12 

0 25% 50% 75% 100% 

0 3 6 9 12 

Revised 3/05/04 14


DAY: 3 



LESSON 21.2, Pp. 382-383 Part 1 

MATERIALS: 

LESSON 

FOCUS: 

CALIFORNIA 

STANDARDS: 

Overhead transparency of squares and several copies per student (Or 

students can draw the squares.) 

Per student: paper to fold; ruler, p. TR27. 

Multiply a Fraction by a Fraction 

Number Sense 2.4 Understand the concept of multiplication and division 

of fractions; 

2.5 Compute and perform simple multiplication and division fractions 

and apply these procedures to solving problems. 

PURPOSE OF 

LESSON: 

LAUNCH: 

Introduce 

students to 

concepts. 

Paper to fold. 

EXPLORE: 

Work with the 


on students 

“doing” 

mathematics. 

PRACTICE: 

Focus on 

Communication 

and 


SUMMARIZE: 

Connect 

purpose to 

activities. 

• To use an area model (array) to help students think about/visualize 

multiplication of a fraction by a fraction; 

• To understand the word “of” as used in 1/4 of 2/3 indicates multiplication 

Learn, P. 382: High Strung. 

• Write problem on the board/overhead. Read with students. 

• Ask students to work in pairs to find a strategy to use to solve. 

• Share and discuss. 

• Model as students work through the Steps. 


Highlight Math Idea, bottom SE P. 382. 

Check P. 383 #2-6, 

Check, P. 383 #1 Discuss. 

Practice & Problem Solving, P. 383 # 22-25, Discuss. 

• Create rectangular arrays to show the products. 

ASSESS P. 383: Discuss. 

• Work with a partner to discuss. (If students do not rotate the square, rotate it 

on the overhead to show the commutative relationship. 

• Relate to whole number multiplication. 

2 × 3 = 3× 2) 

HOMEWORK: Practice & Problem Solving, P. 383: #7-11 

Revised 3/05/04 15


ROUTINES: 

Connect the array to decimal multiplication. 

.2 x .3 = .06 (two-tenths of three tenths equals six hundredths) 

.3 

.2 

Revised 3/05/04 16


Revised 3/05/04 17


Revised 3/05/04 18


DAY: 4 



LESSON 21.2, Pp. 382-383 Part 2 

MATERIALS: 

LESSON 

FOCUS: 

CALIFORNIA 

STANDARDS: 

PURPOSE OF 

LESSON: 

LAUNCH: 

Introduce 

students to 

concepts. 

Overhead square for the teacher and one sheet of squares per student 

Multiply a Fraction by a Fraction 


of fractions; 



• Connect visual work with arrays to the algorithm for multiplying fractions; 

• To look for patterns in the product to generalize a procedure for multiplying 

fractions; 

• To understand that when you multiply fractions less than 1 the product will 

be less than either factor 

Warm-Up: Do together with students. 

Maya brought 2/3 of a cake to school. Use a rectangle to represent what her 

cake might have looked like. Share student ideas. 

Maya shared half of what she had with her 

friends. Use a rectangle to show how much 

of the cake her friends got. Share student 

ideas. 

Jun had 1/2 of a cake. Use a rectangle to 

represent what his cake might have looked like. Share student 

ideas. 

If Jun ate 3/4 of what he had, how much did he eat? Use a 

rectangle to draw a picture of how much he ate. 

EXPLORE: 

Work with the 


on students 

“doing” 

mathematics. 

Practice & Problem Solving, P. 383: #26-28. Check with drawings. 

Revised 3/05/04 19


PRACTICE: Practice & Problem Solving, P. 383: #17-21. Discuss 

Focus on 

Communication 

and 


SUMMARIZE: Practice & Problem Solving, P. 383: #29. Share responses. 

Connect 

purpose to 

activities. 

HOMEWORK: Practice & Problem Solving, P. 383: #12-16. 

ROUTINES: 

Review/Test, P. 390 #1-3, 4, 5, 8, 9. 

Revised 3/05/04 20


DAY: 5 



LESSON 21.3, Pp. 384-385 

MATERIALS: 

LESSON 

FOCUS: 

CALIFORNIA 

STANDARDS: 

PURPOSE OF 

LESSON: 

LAUNCH: 

Introduce 

students to 

concepts. 

Copies of squares for students 

Multiply Fractions and Mixed Numbers 


of fractions; 



To connect work with arrays to multiplying fractions and mixed numbers; To 

use the distributive property to multiply fractions and mixed numbers 

(See Alternative Teaching Strategy on p. 384B); 

To understand how to rename a mixed number as a fraction so you can 

multiply 

Learn, P. 384: Chef’s Garden. Write problem on board/overhead. Discuss. 

Students work with a partner to solve with models. (Provide squares.) 

They share solutions and strategies. 

(If students don’t use the squares, model the problem with them as 

shown at the top of 384.) 


EXPLORE: 

Work with the 


on students 

“doing” 

mathematics. 

Alternative Teaching Strategy, P. 384B 

• Connect Chef’s Garden to Alternative Teaching Strategy: 

• Discuss how using the distributive property can help. 

1/4 × 1 1/3 = 1/4 ( 1 + 1/3) = (1/4 × 1) + (1/4 × 1/3) 

• Connect to whole numbers: 5 × 6 = 5(4 + 2)= (5 × 4) + (5 × 2) 

Can we use our procedure for multiplying fractions to multiply fractions 

and mixed numbers? 

• Work with a partner to try 1/4 × 1 1/3 = 

• Share student solutions and strategies. (Change the mixed number to a 

fraction.) 1/4 × 4/3 = 4/12= 1/3 

Check P. 385: 1-5 

• Check 2-5 by using a second method to make sure your answer makes 

sense. 

• Work with a partner and use squares, the distributive property, or the 

procedure for multiplying fractions to solve these problems. 

Revised 3/05/04 21


PRACTICE: 

Focus on 

Communication 

and 


SUMMARIZE: 

Connect 

purpose to 

activities. 

HOMEWORK: 

ROUTINES: 

Practice & Problem Solving, P. 385 #30-34. 

Share and discuss. 

ASSESS, TE P. 385: WRITE. 

• Share 

Discuss Common Error Alert, margin TE P. 385. Ask students to draw a 

picture to represent changing mixed numbers to fractions. 

Practice & Problem Solving, P. 385 #26-29 Draw models. 

Mixed Review #37-39 

If students have difficulty changing a mixed number to an improper fraction, 

drawing number lines will help them. 

Example: 2 1/3 = ?/3 

0 1 2 2 1 3 

3 

Students mark the thirds to see that 2 1/3 = 7/3 

They can also use their fraction Kit or copies of squares to find the improper 

fraction. 

Revised 3/05/04 22


Revised 3/05/04 23


DAY: 6 



LESSON 21.4, Pp. 386-387 

MATERIALS: 

LESSON 

FOCUS: 

CALIFORNIA 

STANDARDS: 

PURPOSE OF 

LESSON: 

LAUNCH: 

Introduce 

students to 

concepts. 

Multiplying Mixed Numbers 


of fractions; 



Mixed numbers are renamed as improper fractions before multiplying; 

* OPTIONAL: use the GCF to simplify factors when mixed numbers are 

multiplied 

• Write the problem 1/2 × 1 2/3 on the board. 

• Ask students to work with a partner to solve it. Share solution/ strategies. 

(using squares, the distributive property and multiplying fractions by 

changing the mixed number to a fraction.) 

• Write the problem 2 1/4 × 1 2/3 on the board. Ask how this problem differs 

from the previous problem (two mixed numbers) 

EXPLORE: 

Work with the 


on students 

“doing” 

mathematics. 

PRACTICE: 

Focus on 

Communication 

and 


SUMMARIZE: 

Connect 

purpose to 

activities. 

Learn, P. 386: Plan Ahead. Read together. 

• Students work in pairs to solve it. 

• If students do not change both mixed numbers to fractions, ask them to try 

that strategy and see if it works. 


Check, P. 386 #1-3. Do together and discuss. 

Check, P. 387 #4-8 

• Use the multiplying fractions method (Change both mixed numbers to 

fractions and multiply.). 

• If students have difficulty finding the GCF to factor multiplication problems, 

have students do Alternative Teaching Strategy, p. 386B. 

Practice & Problem Solving, P. 387 #25-32. 

• Work with partners. Discuss. Then do #18-20 and discuss. 

ASSESS, TE P. 387: DISCUSS. 

ASSESS, TE P. 387: Write. Give an example. 

OR 

Practice & Problem Solving, P. 387 #33. 

HOMEWORK: Practice & Problem Solving, P. 387 #19-21. 

ROUTINES: 

Mixed Review, P. 387 

Revised 3/05/04 24


DAY: 7 


Module 7: Chapter 22: Divide Fractions 

LESSON 22.1, Pp. 394A-395 

MATERIALS: 

LESSON 

FOCUS: 

CALIFORNIA 

STANDARDS: 

PURPOSE OF 

LESSON: 

LAUNCH: 

Introduce 

students to 

concepts. 

Fraction Kit or Fraction Bars; TR 19 and 20 for Routines 

Hands on Divide Fractions 


of fractions; 



To model division with fractions by using models: Fraction Kit/Fraction Bars 

and number lines (See Routines.); 

To connect division with fractions to division with whole numbers: 

28 ÷ 4 means “How many 4’s are in 28?” 

2 ÷ 1/2 means “How many 1/2’s are in 2?” 


• Students have their Fraction Kits/Bars out. 

• Walk students through Explore, p. 394. 

• Students work with a partner to model each step of the problem. 

How many pieces of ribbon will Kari have? 

• Students work to solve the problem. They share solutions and strategies. 

• Students check their answer by using the 1/3 pieces to see how many Kari 

can cut from two wholes. 

• Ask students to talk to a partner about how to record a number sentence for 

the problem. Record student ideas. 

• Ask which operation (+, -, ×, ÷) makes sense for this problem and why. 

Discuss student ideas. 2 + 1/3? 2-1/3? 2 × 1/3? 2 ÷ 1/3? 

EXPLORE: 

Work with the 


on students 

“doing” 

mathematics. 


Try It a, b, P. 394 

• Write the question for c, d, e (“How many 1/4’s are in 1?”) 

Connect, P. 395. Students model as you discuss. 

Practice, P. 395 #1-3, 5-7 

• Use a model to solve 

• Write the question asked 

• Write the number sentence 

Revised 3/05/04 25


PRACTICE: 

Focus on 

Communication 

and 


SUMMARIZE: 

Connect 

purpose to 

activities. 

HOMEWORK: 

ROUTINES: 

Practice, P. 395 #12 - 14 

ASSESS, P. 395: DISCUSS. 

• Solve with a model. 

• Write the number sentence. 

Practice, P. 395 #9-11. Use models. 

Mixed Review, P. 395. 

Use the number lines on TR 19 and 20 as another model for dividing 

fractions. 

Example: 2 ÷ 2/3 How many 2/3’s are in 2? 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

3 

3 

3 

3 

3 

3 

3 

3 

3 

3 

0 

1 

2 

1 

1 

2 

2 

1 

2 

3 

Revised 3/05/04 26


Revised 3/05/04 27


DAY: 8 



LESSON 22.2, Pp. 396-397 

MATERIALS: 

LESSON 

FOCUS: 

CALIFORNIA 

STANDARDS: 

PURPOSE OF 

LESSON: 

LAUNCH: 

Introduce 

students to 

concepts. 

Practice 22.1 One copy per student 

Fraction Bars 

Reciprocals 


of fractions; 



• To define and find reciprocals 

• To understand reciprocals of any number is 1 divided by that number. 


Learn, P. 397: Working for Peanuts. 

• Write problem on board/overhead. 

EXPLORE: 

Work with the 


on students 

“doing” 

mathematics. 

PRACTICE: 

Focus on 

Communication 

and 


SUMMARIZE: 

Connect 

purpose to 

activities. 


Check, P. 396 # 1. Discuss. 

Check, P. 396 # 2-4 

Practice, P. 397 #38-46. 

• Discuss. 

Practice, P. 397 #22-31. 

• Discuss. 

ASSESS, TE P. 397: Discuss 

ASSESS, TE P. 397: Write. 

HOMEWORK: Practice 22.2 

Mixed Review, P. 397 #47, 49, 50-51 

ROUTINES: 

Revised 3/05/04 28


DAY: 9 



LESSON 22.3, Pp. 398-399 

MATERIALS: 

LESSON 

FOCUS: 

CALIFORNIA 

STANDARDS: 

PURPOSE OF 

LESSON: 

LAUNCH: 

Introduce 

students to 

concepts. 

Divide Whole Numbers by Fractions 


of fractions; 



Multiplication and division are inverse operations; 

To look for patterns in multiplication and division to help students make sense 

to using reciprocals to divide by a fraction 

Learn, P. 398: Slow Motion. 

• Write problem on board/overhead. Discuss. 

• What do we know? (A tortoise walks 1/5 mph.) 

• What do we need to find out? (How many hours will it take the tortoise to 

walk two miles.) 

• What do we need to find out to answer that? (How many 1/5 miles are in 2 

miles.) 

• What number sentence can we write? (2 ÷ 1/5 = ? ) 

Teach, p. 398; Guided Instruction questions to guide discussion. 

• Ask students what patterns they notice. 

(Multiplying by a whole number has the same answer as dividing by its 

reciprocal.) 

See Another Way, SE P. 398. 

1. 6 ÷ 2 = 3 6 × 1/2 = ? 

2. 6 ÷ 1 = 6 6 × 1 = ? 

3. 6 ÷ 1/2 = ? 6 × 2 = ? 

Discussion Point: 

How could you change 6 ÷ 1/4 into a multiplication problem by using the 

reciprocal of 1/4? Would the answer be the same? 

( Dividing by a whole number or fraction has the same answer as multiplying 

by the reciprocal of the divisor.) 

See Step 1-3, P. 398: 

What would happen if the numerator wasn’t 1? 

Does 5 ÷ 2/3 equal 5 × 3/2? 

What question does 5 ÷ 2/3 ask? (How many 2/3’s are in 5?) 

• Connect to number line model: 

• Ask students to draw an open number line and label 0,1,2,3,4,5. 

Revised 3/05/04 29


0 1 2 3 4 5 

• Students divide the number line into thirds. 

0 1 2 3 4 5 

• Circle groups of 2/3 to see how many are in 5. 

1/2 of 2/3 

EXPLORE: 

Work with the 


on students 

“doing” 

mathematics. 

PRACTICE: 

Focus on 

Communication 

and 


SUMMARIZE: 

Connect 

purpose to 

activities. 

0 1 2 3 4 5 

5 ÷ 2/3 = 7 1/2 (7 1/2 two-thirds) 

• What is the reciprocal of 2/3? (3/2) 

5× 3/2 = 5/1 × 3/2 = 15/2 = 7 1/2 

• The answers are the same. 

Check, P. 399 #1. 

• Discuss. 

• Use Fraction Kit/Bars, patterns, open number lines, or reciprocals to solve 

• Check, #2-4. Discuss. Then, do #7-9. 

Practice & Problem Solving, P. 399 #22-33. Discuss. 

“Write”, P. 399: Explain the steps used to divide whole numbers by fractions 

using reciprocals. 

• Chart and add examples. 

HOMEWORK: Practice & Problem Solving, P. 399: 17-21 

ROUTINES: 

Open number lines or TR 19, 20 can be used as models to think about 

division with fractions (understanding the concept) and as tools to solve 

division problems with fractions. (Use the example of an open number line 

provided in Launch.) This model also helps students make sense of the 

answer. 5 ÷ 2/3 = 7 1/2 TWO-THIRDS, not 7 1/2 “wholes”. 

Revised 3/05/04 30


DAY: 10 



LESSON 22.4, Pp. 400A-406 

MATERIALS: 

LESSON 

FOCUS: 

CALIFORNIA 

STANDARDS: 

PURPOSE OF 

LESSON: 

LAUNCH: 

Introduce 

students to 

concepts. 



of fractions; 



Provide practice using the reciprocal of a number to divide by a fraction; 

To extend division with fractions to mixed numbers 

Alternative Teaching Strategy P. 400B 

Learn, P. 400: Buzzzy Work. 

• Read problem with students. 

• Write 2 1/4 ÷ 1/3 on the board. 

• Ask students to estimate the answer. 

• They work with a partner to figure out how to solve using the reciprocal. 

• Share strategies and solutions. (Change 2 1/4 to a fraction.) 

EXPLORE: 

Work with the 


on students 

“doing” 

mathematics. 

PRACTICE: 

Focus on 

Communication 

and 


SUMMARIZE: 

Connect 

purpose to 

activities. 


Emphasize Math Idea. 

Use Reciprocals to Divide, top SE P. 401. 

• Use bullets top margin TE p. 401 to guide discussion. 

Check, P. 401 #1. Discuss. 

• Check, P. 401 #2-6, 10, 13. Discuss. 

Practice & Problem Solving, P. 402 #15, 38, 41-47, 

• Work with partners. 

• Discuss. 

ASSESS, TE P. 403: Discuss. 

ASSESS, TE P. 403: Write. 

HOMEWORK: Practice & Problem Solving, P. 402 #39-40 

Link Up to Reading, P. 403 

ROUTINES: 

Revised 3/05/04 31

Revised 3/05/04 32


Revised 8/20/04 33

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